PSI - Issue 77

Victor Rizov et al. / Procedia Structural Integrity 77 (2026) 397–404 Author name / Structural Integrity Procedia 00 (2026) 000–000

401 5

h

1

∫ 2

= h D lft M b zdz σ , 3

(21)

1

−

2

h

∫ 2

M

b

3 4 1 1 z dz

=

σ

.

(22)

3 D rght

D D

h

−

2

Equations (6), (11), (13) and (20) are used for determining the curvatures and the coordinates of the neutral axis. The integral J is found by Eq. (23). 2 1 B B J J J = + , (23) where 1 B J and 2 B J are solutions in portions, 1 B and 2 B , of the contour, B , that is shown in Fig. 1. Equations (24) and (25) are applied for obtaining of 1 B J and 2 B J . ds x v p x u p J u B y B x B B B ∫ ∂ ∂ + ∂ ∂ − = 1 1 1 1 1 0 1 1 cos α , (24)

  

     

  

1

1

  

     

  

u

v

∂

∂

B ∫ = 2

cos α

J

u

p

p

ds

−

+

2

2

B

B

.

(25)

2

2

B 0 2

2

B

x

y

x

x

∂

∂

1

1

The components of Eqs. (24) and (25) are expressed through the stresses and strains in the left-hand and right-hand sides of section, 3 D , of the beam structure. The integration is carried-out by the MatLab. The strain energy release rate, G , is determined to check-up the integral J . Equation (26) is used for deriving G .

bda dU *

G

=

,

(26)

where * U is the complementary strain energy stored in the beam structure. * U is determined by integrating the specific complementary strain energy in the beam volume. The fact that the strain energy release rate agrees with the integral J is prove for the correctness of the analysis. 3. Parametric analysis In this section of the paper the solution of the integral J is applied for exploring the combined effect of distribution of material properties including the coefficient of thermal expansion along the beam length and the increased temperature on the longitudinal fracture behaviour. The results obtained are reported in the form of various graphs illustrating how the value of the integral J changes when the parameters of the model of the beam with a longitudial crack are varied. The data used are as follows: 0.020 = b m, 0.035 = h m, 0.300 1 = l m, 0.500 2 = l m, 0.200 = a m, 0.010 1 = h m, 0.5 1 = β , 0.6 2 = β and 0.7 3 = β . First, the ratio, 2 1 / t t , is varied in order to explore its efect on the integral J (the latter is presented in normalized form by the formula JL b lft / ).